Linear Free Resolutions and Minimal Multiplicity

Let S = k[x, ,..., x,] be a polynomial ring over a field and let A4 = @,*-a, M, be a finitely generated graded module; in the most interesting case A4 is an ideal of S. For a given natural number p, there is a great interest in the question: Can M be generated by (homogeneous) elements of degree <p? No simple answer, say in terms of the local cohomology of M, is known; but somewhat surprisingly the stronger question: Can the jth syzygy of M be generated by elements of degree Qp + j for all j = 0, l,..., n? does admit a simple response. The following is neither new nor difficult to prove, though it seems not well known: